On local properties of a class of differential operators with weight symbols (Q2719436)

The author proves the theorems on local unsolvability and local solvability of the equation \( P(x,D)u=f \) in the neighborhood of the point \(x_0\). It is assumed that the differential operator of weight order \(m\) is of the form NEWLINE\[NEWLINE P(x,D) = \sum\limits_{\{\alpha\mid \langle\rho,\alpha\rangle\leq m\}} a_\alpha(x)D^\alpha,\quad D_j = \frac 1{i} \frac{\partial}{\partial x_j}, NEWLINE\]NEWLINE where \(\alpha\) is integer multi-index, \( \langle\rho,\alpha\rangle = \rho_1\alpha_1 + \dots + \rho_n\alpha_n\), \( a_\alpha(x)\) are infinitely differentiable functions.